Advection-Dispersion
Model

In order to study
the impact of organic
matter and nutrient loading on food
web dynamics, a 1-D, tidally averaged, advection-dispersion model was
developed. The governing equation of this model is given by

where

x
is distance (m)

t is time (d)

C(x, t) is a vector
of nutrient or organism
concentrations
(in mg m-3)

r(t) is a vector of
source terms that is determined
by the
structure of the food web (mg m-3d-1)

Dispersion
Coefficient and Area

The dispersion
coefficient as a
function of location and river discharge has
been estimated in the Parker River-Plum Island Sound Estuary, MA, from
both dye
studies and salinity transects. In the dye study, an inert dye
(rodamine WT)
was released in the upper Parker River, and its concentration profile
was
measured with a fluorometer at several subsequent high tides.
Parameters in a
function used to model the dispersion coefficient were then adjusted so
that the
dye distribution was accurately reproduced by the advection-dispersion
model. This is illustrated by this
figure,
in
which
measured dye profiles (shown in open symbols) are compared to model
output in
sold lines for tidal cycles 1, 3, and 5.

Under steady state
conditions, the
dispersion coefficient can be estimated
from the following equation

where C(t,x)
is
the concentration of an inert tracer
(i.e. salt) and q(t) is
volumetric flow rate.
Consequently, the
dispersion coefficient can be estimated from salinity profiles. From
salinity
profiles during different river discharge regimes, we have obtained the
following equation to describe dispersion (m2 s-1)
in
the Plum Island Estuary (see
figure)

where x
is distance (m), and q
is volumetric flow rate (m2s-1)

Cross sectional
river area for the
Parker River was measured and fit to the
following simple function

Food Web

The food web model
is based on the
following diagram

which consistes of
autotrophs,
heterotrophs, dissolved inorganic nitrogen
(DIN), and dissolved organic matter (OM) that is represented in both
labile and
refractory pools. The heterotrophs are able to graze the autotrophs as
well as
utilize the DIN and labile OM pools and mineralize organic nitrogen.
The food
web is an example of a "lumped" model, in that the heterotroph pool
consists of organisms ranging in size and taxon from bacteria to
macrozooplankton. The autotroph pool is similarly represented.

Simulation

An animated simulation of the model
output
showing the transient start-up of the food web from initial conditions
can be
viewed by clicking the figure on the right. The
simulation illustrates the dynamics of autotrophs (Auto) being grazed
by
heterotrophs (Hetero) along the length of the estuary (distance in km)
starting
from the head of the river to the ocean. Changes in DIN and labile OM
are due
to both the biotic reactions and mixing with the ocean. Pure mixing is
best
illustrated by the monotonic increase in salinity (Salt), which behaves
as a
conservative tracer in estuaries.

The transient for
the autotrophs last
about 1 wk (1 tidal cycle equals 12.25
hr), while that for the heterotrophs last about 2 wks. However,
predator-prey
oscillations are evident in the down stream portion of the estuary. The
labile
organic matter comes to approximate steady state in only 2 d. Static,
3D images
of the state variables versus time (tidal cycles) and distance (km) can
also be
examined by clicking the thumbnail figures below. Figures were rendered
in
Mathematica.

Autotrophs

Heterotrophs

DIN

Salt

OM labile

OM Refractory

A variation of this
model was used to
examine the autotrophic-heterotrophic nature of estuaries. See
Hopkinson and Vallino (1995) Estuaries,18:
598-621. Also see Vallino, J.J. and Hopkinson, C.S. (1998). Estimation
of Dispersion and Characteristic Mixing Times in Plum Island Sound
Estuary. Estuarine, Coastal and Shelf Science46,
333-350.

Algorithm

Code used to solved
the above problem
is written in Fortran, and the
algorithm employed to solve the 1D PDE is a moving-grid method
developed and
written by Blom and Zegeling (1994), ACM TOMS20:
194-214. (TOMS731).

Steady state
solutions have also been
obtained using a collocation algorithm
to solve the resulting boundary value problem (COLNEW:
Bader and Ascher (1987), SIAM J. Sci. Stat. Comp.8:
483-500).

Although not used in
this particular project, we have found the BACOLR
routine to be very efficient at solving 1D ADV equations (R. Wang, P.
Keast, and P. H. Muir. Algorithm 874: BACOLR-spatial and temporal error
control software for PDEs based on high-order adaptive collocation. ACM Trans.Math.Softw. 34 (3):1-28, 2008.)